KurzfassungDomains and metric spaces are two central tools for the study of denotational semantics in computer science, but are otherwise very different in many fundamental aspects. A construction that tries to establish links between both paradigms is the
space of formal balls, a continuous poset which can be defined for every metric space and that reflects many of its properties. On the other hand, in order to obtain a broader framework for applications and possible connections to domain theory, generalized ultrametric spaces (gums) have been introduced. In this paper, we employ the space of formal balls as a tool for studying these more general metrics by using
concepts and results from domain theory. It turns out that many properties of the
metric can be characterized via its formal-ball space. Furthermore, we can state new results on the topology of gums as well as two new fixed point theorems, which may be compared to the Prieß-Crampe and Ribenboim theorem, and the Banach fixed point theorem, respectively. Deeper insights into the nature of formal-ball spaces are gained by applying methods from category theory. Our results suggest that, while
being a useful tool for the study of gums, the space of formal balls does not provide the hoped-for general connection to domain theory.